
see [1]} and [2]}.
Following Equation (REF ) and assuming the control measure \(m\)  to have a symmetric density function \(f\)  we define the \(\alpha \) -sine transform of \(f\)  as
\(T_\alpha f(t):=\int _0^\infty \left|\sin \left(tx\right)\right|^\alpha f(x)dx=\frac{2\sigma ^\alpha -\tau (2t)}{2^{\alpha +1}\lambda _\alpha }=\frac{\Vert X(2t)-X(0)\Vert _\alpha ^\alpha }{2^{\alpha +1}\lambda _\alpha }.\) 
